7 INTEGRABLE FUNCTIONS AND SETS 121 7. INTEGRABLE FUNCTIONS AND SETS We have seen (13.5.10) that every function/: X -> R satisfies //#(/) ^/j*(/). The function / is said to be integrable (with respect to ju) or ^integrable if M*(/) and /**(/) are finite and equal* Their common value is then called the integral of f with respect to ju, and is written /i(/), or </, /*>, or \fdfji, or [/(*) ^M*)- Clearly every function /e «2rR(X) is integrable and its integral is the value of \i at/, so that our notation is consistent. An integrable function is therefore./zrate almost everywhere (13.6.4), but a bounded function is not necessarily integrable. For example, a constant non- zero function on R, or the continuous function 1/(1 + |jc|), is not integrable (with respect to Lebesgue measure) (cf. (13.20)). In view of the definitions in (13.5), we have the following criterion for integrability: (13.7.1) (i) For a function f: X-»K to be integrable, it is necessary and sufficient that, given any e > 0, there should exist g e £f and he<# such that 9 ^/^ h and /x*(A) — M*(#) => £ (or, equivalently (13.5.3), p*(h — g) g e). (ii) Iffis integrable, there exists a decreasing sequence (hn) of functions belonging to J and an increasing sequence (gn) of functions belonging to $f such that gn^f^ hnfor all n and lim / (i) The condition implies that /t*(A) and fi*(g) are necessarily finite (and therefore so are /**(/) and /**(/)), and that n*(f) — /**(/)• (ii) For each n there exists h'n e £ and g^e^ such that g'n^f^h'n and /*(/) ~ w"1 ^ l**(g'n) ^ K/) ^ M*(%) ^ /*(/) + w"1- The required conditions are then satisfied by taking hn = inf(A'l5 . . . , h'n) and gn — sup(#i, . . . , g£). If /is integrable, then clearly so is any function/! equivalent to/ (13.6), and we have f/i fi?/i = |/rf/x (which we denote also by X/))- It is now clear h°w integrability should be defined for functions defined almost everywhere in X: such a function /is integrable if the functions equivalent to /and defined on the whole of X are integrable. The number /*(/) is also denoted by \fd\i, or /(x rf/<x), or (13.7.2) For a mapping f:X-+Htobe integrable it is necessary and sufficient that, given any e > 0, there exists a function u e «2rR(X) such that this defines an order relation on the set of equivalence classes (with